Theresa May has united the educational establishment in opposition to her plans for new grammar schools, Jeremy Corbyn told the prime minister during a boisterous prime minister’s questions, saying the system would lead to “segregation for the few”.

Using all six questions to press May on her plans to allow selective schools to be set up, the Labour leader said the prime minister had created “unity between Ofsted and teaching unions and former education secretaries on both sides of this house – a true era of unity in education thinking”.

Corbyn presses May over grammar schools at PMQs - Politics live

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Corbyn challenged May to name an educational expert who would back her proposals on grammar schools and selective schools, asking her: “Why does the prime minister want to expand a system that can only let young people down?”

May insisted that new grammars would not lead to poor standards in schools for children who did not get into selective schools.

Corbyn “needs to stop casting his mind back to the 1950s,” May said, claiming that new grammars would improve life chances for 1.25 million children in schools that were failing or in need of improvement.

“He believes in levelling down; we believe in levelling up,” she said. “Can I gently remind the right honourable gentleman, he went to grammar school, I went to a grammar school; it’s what got us where we are today.” However, May conceded: “My side is probably happier about that than his.”

PMQs verdict: Corbyn tests May on grammar schools

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“It’s not about pulling up ladders, it’s providing a ladder for every child,” he said, and quoted the former prime minister David Cameron as saying: “There is a kind of hopelessness about the demand to bring back grammars ... I want the Conservative party to rise above that attitude.”

Corbyn said: “Isn’t he correct to say what we need is investment in all of our schools, a good school for every child, not this selection at the age of 11? The government is heading backwards to failed segregation for the few and second-class schooling for the many.”

The focus on grammar schools drew roars of approval from the Labour backbenches, which have become a rare sound during Corbyn’s year as Labour leader, a period in which the vast majority of the parliamentary party has expressed no confidence in his leadership.

With the results of the Labour leadership race due to be announced next week, after parliament goes into recess, May said the exchange could “be [Corbyn’s] last opportunity to face me across this dispatch box, certainly if his members of parliament have anything to do with it”.

Lucy Powell MP (@LucyMPowell)

Jeremy Corbyn easy win at #PMQs Theresa May has made a serious misjudgement on grammar schools. Her MPs know it.

Corbyn’s performance was praised by several of his critics on Twitter following his exchange with May. The former shadow education secretary Lucy Powell, a key supporter of his leadership rival, Owen Smith, said the debate was an “easy win for Corbyn”, and Jonathan Reynolds, the Labour MP for Stalybridge and Hyde, said the Labour leader was “thrashing the PM”.